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• It is well known fact that Alexander polynomial arises in 3d Chern-Simons theory with superalgebra sl(1|1) [36, 37, 38, 39]. Carefull consideration of the representation theory of superalgebras reveals that Young diagrams enumerate the representations ambiguously. In general for a representation one can align several Young diagrams, that are connected via some relation. If this relation is applied to the representation of the K colored Alexander polynomial AR(q) one obtains the tug-the-hook symmetry. For more careful consideration of this argument see [33].
• The tug-the-hook symmetry together with the rank-level duality explains vanishing of 1,3,5 orders in the loop expansion of the colored Alexander polynomial for arbitrary knot (see Section 5.4). This fact is surely known from the trivalent diagram point of view. All trivalent diagrams at these levels are known and we can establish the vanishing of the corresponding group-factors. However it is not obvious a priori, without calculating the trivalent diagrams explicitly. Also we see that 2,4,6 orders of the loop expansion have the structure compatible with the symmetry.
• The symmetry is tightly connected with the eigenvalue conjecture [26]. One of the possible formulations of the conjecture is the set of quantum Ri-matrices is completely determined by the normalized eigenvalues of the quantum universal Rˇ-matrix. The eigenvalue conjecture has been checked in nu- merous cases and proven in some of them (see Section 2.3 in [30] for a review of checks of the eigenvalue conjecture). In Section 6 we have shown that the symmetry is a corollary of the eigenvalue conjecture. It actually also strengthens our examples (1,2,3). In Section 6 we observe that symmetry follows from con- servation of the R matrices eigenvalues, hence the knot itself does not appear in the proof. Hence verifying it for some knots leads to conclusion that it is true for all knots.
We would like to emphasize that the origin of this symmetry can be described from three different points of view: Lie superalgebras, quantum Lie algebras, and classical Lie algebras. In each case, it follows from the invariance of a certain algebraic structure.
The paper is organized as follows. In Section 2 we review some basic facts about the colored HOMFLY. In Section 3 we define the colored Alexander polynomial and discuss the motivation to study it. In Section 4 we present a new conjectural symmetry of the colored Alexander polynomial. We define the tug-the-hook solutions in Section 5 to study the loop expansion. Full description of the tug-the-hook solutions is presented in Section 5.1 in terms of combinatorics. In Section 5.2 we discuss derivation of the solutions. Section 5.4 is devoted to the explanation of the form of the loop expansion of the colored Alexander with the help of the symmetry. In Section 6 we discuss the connection of the symmetry with the eigenvalue conjecture. The last Section 7 is devoted to possible applications of the tug-the-hook symmetry.
2 The colored HOMFLY polynomial
The HOMFLY polynomial can be arrived in various ways. We mention two of them: • Chern-Simons theory approach. The colored HOMFLY polynomial of a knot K can be obtained as the average of the Wilson loop in Chern-Simons theory with the gauge group SU(N) on S3 [5, 15]. In our notation R stands for a representation of the gauge group and in particular case of SU(N) is enumerated by a Young diagram. K HR (q,a)= trR P exp A , (4) IK CS where Chern-Simons action is given by
κ 2 SCS[A]= tr A ∧ dA + A ∧ A ∧ A . (5) 4π 3 3 ZS The polynomial variables are parameterized as follows:
~ ~ 2πi q = e , a = eN , ~ := . (6) κ + N
One can evaluate (4) in the holomorphic gauge Ax + iAy = 0 [18] and obtain the loop expansion of the
2 HOMFLY polynomial [15] ∞ K K R ~n HR (q,a)= vn,mrn,m . (7) n=0 m ! X X A remarkable fact about this expansion is that the knot and group dependence splits. Knot dependent K parts vn,m are integrals of fields’ averages along the loop. They are known as Vassiliev invariants of knots or R sl invariants of finite-type. Group dependent parts rn,m are called group factors and are traces of N generators Ti. R rn,m ∼ trR T (m) T (m) ...T (m) (8) i1 i2 i2n (m) (m) (m) K i1 i2 in vn,m ∼ dx1 dx2 ... dxn A (x1) A (x2) ...A (x2n) (9) I Z Z D E If one proceeds with several extra transformations of (7) in the holomorphic gauge, one can arrive the Kontsevich integral form [18, 21]. Before turning to this particular form let us review some basic facts about the Kontsevich integral and Lie algebra weight systems [22] (see Chapter 6). Full definition of the Kontsevich integral can be found in [22] (see Chapter 8.2). However, what matters for us is that its values belong to the graded completion of the algebra of unframed chord diagrams Dˆ [22] (see Chapter 4)
∞ Z(K)= V(K)n,m Dn,m. (10) n=0 m X X
We denote Dn,m a chord diagram with n chords and V(K)n,m the coefficient of the chord diagram in the Kontsevich integral. Some examples of the unframed chord diagrams in small degrees are:
D0,1 D2,1 D3,1 D4,1 D4,2 D4,3
The Lie algebra weight system ϕslN is the homomorphism from the algebra of unframed chord diagrams D to the center of the universal enveloping algebra ZU(slN ). The definition of ϕ and the proof of this statement can be found in [22] (see Chapter 6). The mapping ϕslN is clear from examples.
a b* a c*
b b*
b a* a* c
dim slN dim slN ∗ ∗ ∗ ∗ ∗ ϕslN (D2,1)= TaTbTa Tb ϕslN (D3,1)= TaTbTa TcTb Tc a,b a,b,c X=1 X=1 Choosing an irreducible representation of slN identified with the Young diagram R we look at the Lie algebra weight system associated with the representation R
ϕsl R N sl ρR Tr C ϕslN : D −−−→ ZU( N ) −−→ End(V ) −→ (11)
R Linearly extending the action of ϕslN we obtain that the loop expansion of the colored HOMFLY polynomial is the special value of the Kontsevich integral
R K ϕslN (Z(K)) = HR (q,a) (12)
• Reshetikhin-Turaev approach. Knots and links are tightly connected to braids through the Alexan- der’s theorem. It states that every knot or link can be represented as a closure of a braid. In this formalism [11, 12, 13, 14] we construct knot invariants via representations of the braid group. The braid group Bn on n strands has generators σi, where i =1,...,n − 1 with the following relations on them
σi σj = σj σi for |i − j|≥ 2 σi σi+1 σi = σi+1 σi σi+1 for i =1,...,n − 2.
3 For example, the closure of the following braid is the trefoil knot:
σ2 σ1 σ2 σ1 ∈B3
The i-th strand of a n-strand braid is associated with the finite-dimensional module Vi of the quantized universal enveloping algebra Uq(slN ) that can be fully described by the Young diagram Ri. It is well known [19, 20] that with the help of the universal Rˇ-matrix one can construct a representation of the braid group Bn. We consider the quantum deformation parameter q that is not a root of unity: 1 1 ˇ 1 1 Ri := π(σi)= 1 ⊗ ... ⊗ i−1 ⊗ P R⊗ i+2 ⊗ ... ⊗ n ∈ End(VR1 ⊗ ... ⊗ VRn ). (13)
Here P is the permutation operator, namely P (x ⊗ y)= y ⊗ x. The operators Ri satisfy the relations of the braid group Bn:
far commutativity property RiRj = Rj Ri for |i − j|≥ 2 braiding property RiRi+1Ri = Ri+1RiRi+1 for i =1,...,n − 2. Graphically the braiding property is the third Reidemeister move, while algebraically it is the quantum Yang- Baxter equation on the quantum R-matrices.
K Let β ∈ Bn be a braid whose closure gives the knot K. In case of knots we need only one Young dia- gram R to describle n equivalent copies of the module VR. The closure operation corresponds to taking quantum trace and we obtain the colored HOMFLY polynomial
K K HR (q,a)= qtr ⊗n π(β ) . (14) VR
⊗n For an element z ∈ End(VR ) the quantum trace is defined as follows:
qtr ⊗n (z) := tr ⊗n (K2ρ z) , VR VR where ρ is a half-sum of positive roots. In terms of simple roots αi
N−1 N−1 ni (αi , H) 2ρ = ni αi,K2ρ := Ki ,Ki := q . i=1 i=1 X Y ⊗n To compute the colored HOMFLY it is convinient to decompose VR into irreducible components:
⊗n VR = MQ ⊗ VQ. (15) QM⊢n|R| Here the sum runs over Young diagrams Q that appear in the tensor product according to the Littlewood- Richardson rule. The symbol VQ stands for the module enumerated by the Young diagram Q, while the symbol MQ stands for the space of the highest weight vector corresponding to Q. The dimension of the space MQ is called the multiplicity of the representation VQ. For example:
⊗3 V[1] = V[3] ⊕ V[2,1] ⊕ V[2,1] ⊕ V[1,1,1].
In this case M[2,1] is a two dimensional vector space.
A crucial property of quantum Ri-matrices is that they act on the modules VQ by identity:
1 Ri = (Ri)MQ ⊗ VQ . (16) QM⊢n|R| Using this fact we can simplify the expression for the colored HOMFLY. Finally it looks like a character decomposition:
K K K K ∗ ⊗n HR = qtr π(β ) = tr MQ π β · tr VQ (K2ρ)= σQ β · sQ, (17) VR Q ⊢n|R| Q ⊢n|R| X X
4 ∗ where sQ is a quantum dimension. The quantum dimension is defined to be Schur polynomial sQ at the special point [16]: ∗ s := sQ(x1,...,xN ) . (18) Q N+1−2i xi=q
K It is worth to note that with such a definition the colored HOMFLY polynomial HR is not actually a polynomial in the variable q, but a rational function. To get a polynomial one has to normalize it by its value on the unknot, which evaluates to the quantum dimension. From now on we will work with the normalized HOMFLY polynomial: K K HR ∗ HR := , HR (q,a)= sR(q,a) (19) HR
3 The colored Alexander polynomial
Considering the normalized colored HOMFLY polynomial we can set a =1 and obtain a colored knot invariant which in the fundamental representation R = [1] coincides with the famous Alexander polynomial [17]. One can define the colored Alexander polynomial as follows:
K K AR(q) := HR(q,a = 1). (20)
The study of the colored Alexander polynomial can improve our understanding of a more complicated case of the colored HOMFLY polynomial. Our interest in the colored Alexander polynomial is also supported by the fact that it has several remarkable properties:
• As a function of a representation R and the quantum deformation parameter q the colored Alexander polynomial respects the 1-hook scaling property:
K K |R| L AR(q)= A[1](q ), where R = [r, 1 ]. (21)
This property was conjectured in [2] and proven for torus knots in [3]. Also it was shown in [4] that the eigenvalue conjecture implies this property. An example of a 1-hook Young diagram looks like:
R = [8, 13].
• There is the connection with the Kadomtsev-Petviashvili hierarchy [32]. The 1-hook scaling property induces so called Alexander equations [1]
L Xn,m([r, 1 ]) = 0. (22)
These equations are homogeneous polynomials in Casimir invariants C1(R), C2(R),...,Cn(R). The graded vector space of solutions of the Alexander equations, that contain monomials with even number of variables, and a ring of polynomials, generated by the dispersion relations of 1-soliton KP τ-function, appear to be the same vector spaces [1]. We provide some examples for small orders of n:
The dispersion relations of 1-soliton solutions can be obtained by replacing Hirota derivatives Di with ki in the KP equations in the Hirota form. Since the KP hierarchy is well studied, this correspondence gives a hope that other interesting properties of the colored Alexander polynomial can be found.
5 Alexander equations Dispersion relations KP equations
4 2 4 2 4 2 X4,1 = C1 − 4C1C3 +3C2 k1 − 4k1k3 +3k2 =0 D1 − 4D1D3 +3D2 τ ⊗ τ =0 3 3 3 X5,1 = C2C1 − 3C4C1 +2C2C3 k2k1 − 3k4k1 +2k2k3 =0 D2D1 − 3D4D1 +2D 2D3 τ ⊗ τ =0 4 2 X5,2 = C1 C1 − 4C1C3 +3C2 Table 1: Single hook solutions and relation with KP.
4 The tug-the-hook symmetry
Let us present a new conjectural symmetry of the colored Alexander polynomial. To define the symmetry we introduce the Frobenius notation [31] for the Young diagram R = [R1, R2,...,Rl(R)]. This notation reflect a nice graphical interpretation of the action on Young diagrams:
T αi := Ri − i +1 and βi := Ri − i +1 (23)
R = (α1,...,αr | β1,...,βr)
where α1 > α2 >...>αr > 0, β1 >β2 >...>βr > 0.
The action on Young diagrams is defined as follows:
Tǫ(R) = (α1 + ǫ,...,αr + ǫ | β1 − ǫ,...,βr − ǫ), (24)
where ǫ is an integer and Tǫ(R) is still a Young diagram. Note that all hooks are translated by ǫ. For example:
R = (8, 6, 4 | 7, 4, 1) T−1(R) = (7, 5, 3 | 8, 5, 2) T−2(R) = (6, 4, 2 | 9, 6, 3) T−3(R) = (5, 3, 1 | 10, 7, 4)
The tug-the-hook symmetry claims that for any possible ǫ
K K AR(q)= ATǫ(R)(q) (25)
This new symmetry partially generalizes the 1-hook scaling property (21). Note that the 1-hook scaling property claims that the Alexander polynomial colored with a 1-hook diagram depends only on the size of the diagram. This statement can be obtained from (25) by substituting R as a 1-hook diagram. However, the 1-hook scaling property contains additional information, namely r.h.s of (21). The counterpart for this scaling property is not found yet for the tug-the-hook symmetry.
It is worth to mention that the 1-hook scaling property and the tug-the-hook symmetry fails for the en- tire HOMFLY polynomial, for example:
K K H[2](q,a) 6= H[1,1](q,a), (26)
K K 2 H[2](q,a) 6= H[1](q ,a). (27) It would be interesting to find the extensions of these properties to colored HOMFLY polynomials.
6 5 The tug-the-hook solutions
In this section we define the tug-the-hook solutions. If our conjecture about the tug-the-hook symmetry is true, then the group-factors should be linear combinations of the tug-the-hook solutions. The study of the basis of the tug-the-hook solutions provides us an argument in support of existence of the symmetry.
As the specialization of the HOMFLY invariant the Alexander polynomial inherits its perturbative expansion (7) ∞ K ~ K R ~n AR q = e = vn,mrn,m . (28) N=0 n=0 m ! X X R As we discussed in the introduction, group factors rn,m are the images of the Lie algebra weight system R associated with the representation R. From the map sequence (11) it follows that rn,m can be expressed through the eigenvalues of the Casimir operators as they form a basis in the center of the universal enveloping algebra R rn,m = α∆,mC∆(R), (29) |∆X|≤n where we label the monomials of Ck by the Young diagrams in accordance with the following notation: l(∆)
C∆ = C∆i . (30) i=1 Y Equality (25) holds at all orders of ~ in expansion (28). Moreover, since Vassiliev invariants depend only on a knot, we get the following property for the Alexander group factors:
R Tǫ(R) rn,m = rn,m . (31) N=0 N=0
Now let us move to the more general problem. We consider linear combinations of monomials C∆ that respect the tug-the-hook symmetry. We call these combinations tug-the-hook solutions Yn,m. Indeed, R group factors rn,m are the special case of these solutions. It turns out that the Casimir invariants Ck transform simply under the action of the tug-the-hook symmetry. In the next section 5.2 we derive explicit formulas and show that tug-the-hook solutions are homogeneous polynomials in the Casimir invariants Ck (m) Yn,m(R) := ξ∆ C∆(R), (32) |∆X|=n
Yn,m(R)= Yn,m(Tǫ(R)) (33) where we enumerate by m the independent solutions on the fixed level n. Let us denote the subspace spanned by tug-the-hook solutions order n by Yn. Then we have m =1,..., dim Yn.
Yn := Span Yn,m (34) m ! M We define a graded space of all solutions Y := Yn (35) n M In this work we are interested in the space of tug-the-hook solutions Y. More precisely we aim to clarify two topics:
• The number of independent solutions dim Yn.
• The explicit form of coefficients ξ∆. Y R The main motivation to study is that group-factors rn,m of the Alexander polynomial are linear combina- tions of the tug-the-hook solutions:
R rn,m = λk,lk Yk,lk ∈ Yk (36) k n l k n X≤ Xk M≤
The coefficients λk,lk are unknown and we leave this problem for future studies. These coefficients can be computed using some other methods and we provide an example up to the order 6 (62).
7 5.1 Linear and multiplicative basis of the tug-the-hook solutions In this section we provide the description of the explicit form the tug-the-hook solutions that we have defined in the previous section. The tug-the-hook solutions are turned to be solutions of the special linear systems of equations. These linear systems will be formulated in section 5.2. In this section we also show that the construction of the full set of the tug-the-hook solution Y is reduced to pure combinatorics.
We state that the number of independent solutions on the given level n is:
dim Yn = p(n) − p(n − 1) (37) where p(n) is the number of Young diagrams with n boxes. For small degrees it looks like
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 (38) dim Yn 1 1 1 2 2 4 4 7 8 12 14 21 24 34 41
Let us change the notation to simplify the formulas. From now on, we use YΛ for Yn,m, where Λ is the Young diagram. Then formula (32) becomes: Λ YΛ = ξ∆C∆. (39) |∆X|=|Λ|
Two linear bases of Yn are constructed. Here we list the properties of the first one:
1. YΛ is labeled by Young diagrams Λ = [Λ1, Λ2,..., Λr], where Λ1 =Λ2 ≥ Λ3 ... ≥ Λr.
This fact is in accordance with the formula for dimensions (37). Indeed, basis diagrams in Yn do not contain diagrams that can be obtained by gluing one additional box to the first row of any diagram on the level n − 1. Note that n =1 is the exception.
We list basis diagrams up to the 8-th level:
Y1 : : Y[1]
Y2 : : Y[1,1]
Y3 : : Y[1,1,1]
Y4 : : Y[2,2] : Y[1,1,1,1]
Y5 : : Y[2,2,1] : Y[1,1,1,1,1]
Y6 : : Y[3,3] : Y[2,2,2] : Y[2,2,1,1] : Y[1,1,1,1,1,1]
Y7 : : Y[3,3,1] : Y[2,2,2,1] : Y[2,2,1,1,1] : Y[1,1,1,1,1,1,1]
8 Y8 : : Y[4,4] : Y[3,3,2] : Y[2,2,2,2] : Y[3,3,1,1]
: Y[2,2,2,1,1] : Y[2,2,1,1,1,1] : Y[1,1,1,1,1,1,1,1]
2. The sum in formula (39) is restricted to the diagrams ∆ with properties l(∆) = l(Λ) and ∆ ≥ Λ, where ≥ means a lexicographical order. Λ YΛ = ξ∆C∆ (40) |∆|=|Λ| l(∆)=Xl(Λ) ∆≥Λ
Λ 3. The coefficients ξ∆ have the following structure:
Λ Λ ∆1−Λ1 µ∆ ξ∆ = (−1) (41) i ∆i! Λ Q where µ∆ is the integer coefficient and its description will be given in section 5.2. For illustrative purpose we list linear basis explicitly up to the 8-th level: Y 1 : Y[1] = C[1] Y 2 : Y[1,1] = C[1,1] Y 3 : Y[1,1,1] = C[1,1,1] Y 1 2 4 : Y[2,2] = 2!2! C[2,2] − 3! C[3,1]
Y[1,1,1,1] = C[1,1,1,1] Y 1 2 5 : Y[2,2,1] = 2!2! C[2,2,1] − 3! C[3,1,1]
Y[1,1,1,1,1] = C[1,1,1,1,1] Y 1 2 2 6 : Y[3,3] = 3!3! C[3,3] − 4!2! C[4,2] + 5! C[5,1] 1 3 3 Y[2,2,2] = 2!2!2! C[2,2,2] − 3!2! C[3,2,1] + 4! C[4,1,1] 1 2 Y[2,2,1,1] = 2!2! C[2,2,1,1] − 3! C[3,1,1,1]
Y[1,1,1,1,1,1] = C[1,1,1,1,1,1] Y 1 2 2 7 : Y[3,3,1] = 3!3! C[3,3,1] − 4!2! C[4,2,1] + 5! C[5,1,1] 1 3 3 Y[2,2,2,1] = 2!2!2! C[2,2,2,1] − 3!2! C[3,2,1,1] + 4! C[4,1,1,1] 1 2 Y[2,2,1,1,1] = 2!2! C[2,2,1,1,1] − 3! C[3,1,1,1,1]
Y[1,1,1,1,1,1,1] = C[1,1,1,1,1,1,1] Y 1 2 2 2 8 : Y[4,4] = 4!4! C[4,4] − 5!3! C[5,3] + 6!2! C[6,2] − 7! C[7,1] 1 2 1 5 5 Y[3,3,2] = 3!3!2! C[3,3,2] − 4!2!2! C[4,2,2] − 4!3! C[4,3,1] + 5!2! C[5,2,1] − 6! C[6,1,1] 1 2 2 Y[3,3,1,1] = 3!3! C[3,3,1,1] − 4!2! C[4,2,1,1] + 5! C[5,1,1,1] 1 4 8 8 Y[2,2,2,2] = 2!2!2!2! C[2,2,2,2] − 3!2!2! C[3,2,2,1] + 4!2! C[4,2,1,1] − 5! C[5,1,1,1] 1 3 3 Y[2,2,2,1,1] = 2!2!2! C[2,2,2,1,1] − 3!2! C[3,2,1,1,1] + 4! C[4,1,1,1,1] 1 2 Y[2,2,1,1,1,1] = 2!2! C[2,2,1,1,1,1] − 3! C[3,1,1,1,1,1]
Y[1,1,1,1,1,1,1,1] = C[1,1,1,1,1,1,1,1]
9 Now let us discuss the second linear basis. The set Y has the structure of graded algebra of polynomials
Yn × Ym −→ Yn+m. (42)
We present a multiplicative basis in Y. This multiplicative basis is the subset of the first linear basis and enumerated by Young diagrams of three types
• λ = [λ1, λ2,...,λr], where λ1 = λ2 > λ3 >...>λr ≥ 2
• λ = [λ1, λ2,...,λr], where λ1 = λ2 = λ3 >...>λr ≥ 2
• λ = [1]
Basis elements of the second linear basis are products of the multiplicative basis elements. Considering basis solutions up to the 8-th level we have the following multiplicative basis elements:
Y[1], Y[2,2], Y[3,3], Y[2,2,2], Y[3,3,2], Y[4,4]. (43)
Using these elements we produce the second linear basis up to the 8-th level:
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8
2 3 4 5 6 7 8 Y[1] Y[1] Y[1] Y[1] Y[1] Y[1] Y[1] Y[1] 2 3 4 Y[2,2] Y[2,2]Y[1] Y[2,2]Y[1] Y[2,2]Y[1] Y[2,2]Y[1] 2 Y[2,2,2] Y[2,2,2]Y[1] Y[2,2,2]Y[1] 2 Y[3,3] Y[3,3]Y[1] Y[3,3]Y[1] 2 Y[2,2]
Y[3,3,2]
Y[4,4]
In this basis we observe nicely looking structure that can be generalized to an arbitrary level n. Considering Young diagrams that enumerate the first linear basis we "сut" them into pieces. The pieces are Young diagrams that stand for the multiplicative basis. We "cut" a diagram until it does not contain rows of equal length in the middle:
(44)
Using formulas (40) and (41) we obtain explicit expressions for the multiplicative basis elements and thus for the linear basis element.
10 5.2 Derivation of the tug-the-hook solutions In this part of the paper we present a method to explicitly derive the tug-the-hook solutions. For this reason let us choose a basis in the the center of the universal enveloping algebra ZU(slN ) in the form:
l(R) n n Cn(R)= (Ri − i +1/2) − (−i +1/2) . (45) i=1 X This basis is distinguished by the following facts. The corresponding Hurwitz partition function [25] becomes a KP τ-function [23] and in terms of the Hurwitz partition function, this basis corresponds to the completed cycles and establishes a correspondence with the Gromov-Witten theory [24]. Further calculations look simplier in the Frobenius notation (23). Let us denote the number of hooks in the Young diagram R as h(R), then h(R) n n Cn(R)= (αk − 1/2) − (−βk +1/2) . (46) k X=1 The tug-the-hook symmetry acts on the Casimir invariants as the translation:
n−1 n C (T (R)) = (α + ǫ − 1/2)n − (−β − (−ǫ)+1/2)n = ǫp C (R). (47) n ǫ i i p n−p i p=0 X X This formula allows us to study how monomials of the Casimir invariants transform under the action of the tug-the-hook symmetry
n l(∆) T p ∆i C∆( ǫ(R)) = ǫ C∆i−ki (R) . (48) ki p=0 k ... k p i=1 X 1+ X+ l(∆)= Y Let us consider the symmetry equation on the level n (33) and find constraints on coefficients ξ∆. By definition tug-the-hook solutions are invariant under the action of the symmetry. Monomials of the Casimir invariants turn to the polynomials in variable ǫ (48). We require the vanishing of coefficients in front of all positive powers of ǫ. By straightforward algebraic manipulations we get that the independent constraints come only from the coefficients of ǫ1 and the higher constraints are linear combinations of them. From this fact we get that the sum (32) contains only Young diagrams on the level n. So, the coefficient of ǫ1 has to be zero:
l(∆) ∆ 0= ξ j C C . (49) ∆ 1 ∆j −1 ∆i j=1 i j |∆X|=n X Y6= In (49) the sum is over the diagrams on the |∆|-th level. This sum is actually a linear combination of the Casimir invariants on the (|∆|− 1)-th level. We represent this equation in matrix form
M(n) δ,∆ξ∆ =0, (50) |∆X|=n M(n) where |δ| = n − 1. The matrix δ,∆ has two indices δ, ∆ that are Young diagrams. Diagrams δ are ordered into sets of the increasing length l(δ). In the each set of fixed l(δ) diagrams are in the lexicographical order. Diagrams ∆ are divided into two subsets. The first subset contains diagrams with unequal length of the first and second rows. The second subset contains the remaining diagrams. The diagrams in each subset are ordered in the same manner as diagrams δ. Let us give an example M(6):
11 ∆ δ
6 1 5 2 1 1 4 3 1 2 · 1 4 2 1 1 3 2 1 3 · 1